3 problems

Let $X\subset [N]$ be a subset and let $\epsilon>0$ be real. Assume that, for every prime $p\geq\epsilon^{-1}$, the reduction $X\pmod p$ occupies at most $0.99p$ residue classes. F…

Let $X\subset\mathbb{Z}$ be a subset and let $\epsilon>0$ be real. Assume that, for every prime $p\geq\epsilon^{-1}$, the reduction $X\pmod p$ occupies at most $0.99p$ residue clas…

Inverse sieve conjecture. At least one of the following holds: $|X|\ll_{\epsilon}N^{\epsilon}$, or there exists a polynomial $f(x)\in\mathbb{Q}[x]$ of degree $d\in[2,C]$ and height…